You do NOT need to register with a political party in order to vote in the New York State General election.
HOWEVER
You DO need to register with a political party to vote in the New York State Primary elections. Primary elections take place prior to the general election in order to narrow down the number of candidates running in each party and you can only vote in your registered party's primary (i.e. if you are registered as a Republican, you cannot vote in the Democratic Party primary)